Awesome! You are doing a good job of Chemistry as well as Maths! Because water is made up of $$2 \, \mathrm{amu}$$ of Hydrogen and $$16 \, \mathrm{amu}$$ of Oxygen, the ratio is $$2:16 = 1:8$$.
Do you recall our discussion in the previous module, where we said that if you take $$9 \, \mathrm{gm}$$ of water, you will get $$1 \, \mathrm{gm}$$ of Hydrogen and $$8 \, \mathrm{gm}$$ of Oxygen? We had said that this is because water is made up of Hydrogen and Oxygen in the ratio $$1:8$$?
You can now see why the ratio is $$1:8$$. It is because water is made up of $$2$$ H atoms and $$1$$ O atom. If you use the mass of these atoms, the ratio is $$1:8$$, as you yourself just calculated!